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-Every President of the 20th century first elected in a year ending in 8 either lost re-election, declined to run, or served less than two elected terms.

Examples:
1908: William H. Taft elected--Defeated in 1912
1928: Herbert Hoover elected--defeated in 1932
1948: Harry S. Truman elected, lost in the primaries and declined to run in 1952. He served nearly 4 years of FDR's term, but wasn't elected President in 1944; Thus, he did not complete two full elected terms.
1968: Richard M Nixon -- re-elected in 1972, but resigned in 1974, thus he did not complete two full elected terms.
1988: George H.W. Bush elected; he loses in 1992

Now in every year ending in 4 in the 20th and 21st century, the incumbent President won, and half of those incumbents were in office only due to the death of the President before them:

Two years ending in 6 were won by narrow margins; two other years ending in 6 were landslides:

1916: Woodrow Wilson--Barely re-elected, getting only 49.2% of the popular vote
1936: Franklin Roosevelt re-elected in a landslide, winning 60.8% of the popular vote.
1956: Dwight D. Eisenhower re-elected in a landslide, winning 57.4% of the popular vote
1976: Jimmy Carter barely elected, warning 50.1% of the popular vote
1996: Bill Clinton barely re-elected, winning only 49.2% of the popular vote

Every President elected or re-elected in a year ending in 0 has either died in office, nearly died in office, or had a national catastrophe or embarrassment happen under their watch: